L(s) = 1 | + (0.557 + 0.405i)2-s + (−0.824 − 0.367i)3-s + (−0.471 − 1.44i)4-s + (−1.85 + 3.21i)5-s + (−0.311 − 0.538i)6-s + (0.510 − 0.567i)7-s + (0.750 − 2.31i)8-s + (−1.46 − 1.62i)9-s + (−2.33 + 1.03i)10-s + (4.02 + 0.855i)11-s + (−0.143 + 1.36i)12-s + (0.304 + 2.89i)13-s + (0.514 − 0.109i)14-s + (2.70 − 1.96i)15-s + (−1.11 + 0.807i)16-s + (−1.27 + 0.272i)17-s + ⋯ |
L(s) = 1 | + (0.394 + 0.286i)2-s + (−0.475 − 0.211i)3-s + (−0.235 − 0.724i)4-s + (−0.829 + 1.43i)5-s + (−0.127 − 0.219i)6-s + (0.193 − 0.214i)7-s + (0.265 − 0.817i)8-s + (−0.487 − 0.541i)9-s + (−0.738 + 0.328i)10-s + (1.21 + 0.257i)11-s + (−0.0415 + 0.394i)12-s + (0.0843 + 0.802i)13-s + (0.137 − 0.0292i)14-s + (0.698 − 0.507i)15-s + (−0.277 + 0.201i)16-s + (−0.310 + 0.0659i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.680268 + 0.0483898i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.680268 + 0.0483898i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 + (1.63 + 5.32i)T \) |
good | 2 | \( 1 + (-0.557 - 0.405i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.824 + 0.367i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (1.85 - 3.21i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.510 + 0.567i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-4.02 - 0.855i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.304 - 2.89i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.27 - 0.272i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.397 + 3.77i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.01 - 3.12i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-5.20 - 9.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.690 + 0.307i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.748 + 7.11i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-0.708 + 0.514i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.39 - 2.65i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.847 - 0.377i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + (-1.04 + 1.80i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.17 + 5.74i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-5.53 - 1.17i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-13.7 + 2.93i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (12.8 - 5.73i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.37 + 4.21i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.63 + 5.01i)T + (-78.4 + 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.06848893303481188828549091183, −15.36316272134762891054513202424, −14.73653459317990643314113371739, −13.75641095488656317955596332716, −11.74588415655120802281211696997, −11.06366485160697674112665060806, −9.418449148627935586510313399827, −7.09706963470709097559592763467, −6.24146006614560584595613256991, −4.02894472949931429174626488087,
4.01676555322226381565934810542, 5.32364987605313280391396869349, 8.003119174704505119944117621173, 8.889321713410093184860549490613, 11.18779269540851023814194574804, 12.10140728392505623702518237012, 12.91816189161627770017170533013, 14.41533391248187578855309062397, 16.26111349419193082015783042272, 16.66920327922827264686377880385